Nielsen‐Schreier and the Axiom of Choice

Abstract

The Nielsen‐Schreier theorem asserts that subgroups of free groups are free. In the first section we show that this theorem does not follow from the Linear Ordering Principle, thus strengthening the fact that it implies the Axiom of Choice for families of finite sets. In the second section, we show that a stronger variant of the Nielsen‐Schreier theorem implies the Axiom of Choice.